Typical Peak Sidelobe Level of Binary Sequences
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Alon, N., S. Litsyn, and A. Shpunt. “Typical Peak Sidelobe Level
of Binary Sequences.” Information Theory, IEEE Transactions
On 56.1 (2010) : 545-554. Copyright © 2010, IEEE
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http://dx.doi.org/10.1109/tit.2009.2034803
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Institute of Electrical and Electronics Engineers
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 56, NO. 1, JANUARY 2010
545
Typical Peak Sidelobe Level of Binary Sequences
Noga Alon, Simon Litsyn, Senior Member, IEEE, and Alexander Shpunt
Abstract—For a binary sequence Sn = fsi : i = 1; 2; . . . ; ng 2
f61gn ; n > 1, the peak sidelobe level (PSL) is defined as
n0k
M (Sn ) = k=1;max
ss
:
2;...;n01 i=1 i i+k
It is shown that the distribution of M (Sn ) is strongly concentrated,
and asymptotically almost surely
p
M (S )
(Sn ) = p n 2 [1 0 o(1); 2]:
n ln n
Explicit bounds for the number of sequences outside this range are
provided. This improves on the best earlier known result due to
1 ); 2], and settles
Moon and Moser that the typical (Sn ) 2 [o( pln
n
to the affirmative the conjecture of Dmitriev and Jedwab on the
growth rate of the typical peak sidelobe. Finally, it is shown
p that
modulo some natural conjecture, the typical (Sn ) equals 2.
Index Terms—Aperiodic autocorrelation, concentration, peak
sidelobe level (PSL), random binary sequences autocorrelation,
second moment method.
I. INTRODUCTION AND DEFINITIONS
L
ET
, where
. Define
Bernasconi model [3], which is fascinating for the fact of being
completely deterministic, but nevertheless having highly disordered ground states (sequences with the largest merit factor)
and thus possessing similarities to the real glasses, with many
features of a glass transition exhibited [3], [11].
Study of the problem started in the 1950s. Special attention
has been given to the estimation of typical PSL. Since this is our
central interest in this paper, let us mention several relevant results. Moon and Moser [19] proved that for almost all sequences
for any
. Mercer [18] showed that
Apparently, the suggested approach also allows proving that this
bound is indeed true for most of the sequences (see comments
in [18, bottom of p. 670]). Dmitriev and Jedwab [4] conjectured
and provided experimental evidence that the typical PSL be. The same was presumed without proof
haves as
by Ein-Dor, Kanter, and Kinzel [5].
In this paper, we prove that indeed, for almost all binary seof length
. Moreover, it is
quences
shown that asymptotically almost surely
(1)
The peak sidelobe level (PSL)
Let
of a sequence
, is
stand for the optimal value of the PSL over the set
Binary sequences with low PSL are important for synchronization, communications, and radar pulse design, see, e.g.,
[6], [7], [12], [21], [22]. In theoretical physics, study of the
PSL landscape was introduced by Bernasconi via the so-called
Manuscript received August 22, 2008; revised February 04, 2009. Current
version published December 23, 2009. This work was supported in part by a
USA-Israeli BSF grant, by a grant from the Israel Science Foundation, and by an
ERC advanced grant. The work of S. Litsyn was supported in part by ISF under
Grant 1177/06. The work of A. Shpunt was supported in part by the Di Capua
MIT Graduate Fellowship.
N. Alon is with the Schools of Mathematics and Computer Science, Sackler
Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv, Ramat-Aviv 69978,
Israel (e-mail: nogaa@math.tau.ac.il).
S. Litsyn is with the Department of Electrical Engineering–Systems, Tel-Aviv
University, Tel-Aviv, Ramat-Aviv 69978, Israel (e-mail: litsyn@eng.tau.ac.i)l.
A. Shpunt is with the Department of Physics, the Massachusetts Institute of
Technology, Cambridge, MA 02139 USA (e-mail: ashpunt@mit.edu).
Communicated by G. Gong, Associate Editor for Sequences.
Digital Object Identifier 10.1109/TIT.2009.2034803
The results of the paper have application to another problem
related to estimation of the “level of randomness” of finite se. Mauduit and Sárkózy [17] introduced the corquences from
relation measure of order , which is defined for a sequence
as
In other words, this is just the maximum of the absolute value of
the mutual correlation of continuous runs of vector’s entries.
In [1], Alon, Kohayakawa, Mauduit, Morreira, and Rödl showed
that asymptotically almost surely
(2)
, we conclude that any lower
Noticing that
bound on the typical
is a lower bound for the typical
as well. Therefore, our results (slightly) improve the
lower bound on
0018-9448/$26.00 © 2009 IEEE
in (2) to
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 56, NO. 1, JANUARY 2010
The same improvement can be easily achieved for any using
the method in the paper.
The paper proceeds as follows. In the next short section we
sketch a quick proof, based on the approach in [1], that for
almost all sequences
. We then proceed to give a detailed
analysis that provides a somewhat better control of the error
terms in the above estimates. In Section III, we recall a theorem
due to Moon and Moser [19] for the number of sequences
such that
for any
and
. We then provide estimates for binomial coefficients allowing approximation of the Moon–Moser
formula by tails of the Gaussian distribution with vanishing
error. Section IV is devoted to proving the upper bound in
(1). To do so, we relate, via the Moon–Moser theorem, the
with
number of sequences
to certain binomial sums. Accurate estimates using bounds developed in Section III allow to establish the sought inequality.
Section V derives a lower bound for the number of sequences
with
, by looking only at
. This allows to consider
autocorrelations with shifts
as a collection of linear forms
and variables
.
with coefficients
We then apply the Azuma inequality to show concentration of
, and the results of Sections IV and V to accurately lo, and thus establish the lower bound in
cate the mean of
(1). In Section VI, we argue that modulo a plausible conjecture,
and using the Azuma inequality, for most
are random and independent members of
. A detailed
proof of this simple yet somewhat surprising fact appears in
, let
deSection V. For each
note the indicator random variable whose value is if the event
(which we denote here by
)
occurs, and is otherwise. Our objective is to show that asymptotically almost surely, the sum
is positive. By standard estimates, for each fixed , the probaoccurs is bigger than, say,
(for every fixed
bility that
and all sufficiently large .) This means that the expectaof is at least
. The crucial point is that since
tion
the indicator random variables
are pairwise independent, the
of is the sum of variances of the variables
variance
, and is thus smaller than the expectation of . Therefore,
is zero is at
by Chebyshev’s inequality, the probability that
most
, implying that
asymptotically almost surely is positive, as needed.
The detailed proof, with a more careful treatment of the error
terms, is given in the next sections. We present The proof of the
lower bound that we present in Section V is slightly different
than the one indicated above, as it seems interesting to describe
an alternative approach which derives the bound by combining
the pairwise independence of the random variables described
above with Azuma’s inequality.
We attempted to make the paper as self-contained as possible.
To achieve this, we have included several sketchy proofs of relevant results from other papers conveying ideas of importance
for our presentation.
Theorem 3.1 (Moon–Moser [19]): For
and
II. A QUICK SKETCH
In this short section, we sketch a quick proof that for almost
all sequences
The upper bound is simple (see, e.g., [18]); for each fixed ,
is easily seen to be a sum of
indepenthe sum
and with
dent random variables, each attaining the values
equal probability. It thus follows by standard estimates (cf., for
example, [2, Corollary A.1.2]) that for a random sequence ,
the probability that
is at most
. Thus, for
, where
is
(for
arbitrarily small, this probability is much smaller than
all sufficiently large ), and it thus follows that with high probaare smaller than
,
bility, all numbers
providing the required upper bound.
To prove the lower bound, we consider only values of sat. It turns out that for
which are both bigger
isfying
, the
products
than
III. AUXILIARY RESULTS
denote the number of sequences , such that
. Throughout, we shall adopt the convention that
equals if is not an integer and
the binomial coefficient
if
.
Let
Proof: See a sketch in the Appendix.
Note that one of the consequences of Theorem 3.1 is that
.
In the derivation of our bounds, we will need the following
estimates for binomial coefficients.
Lemma 3.2: For
is an integer
, and all , such that
(3)
(4)
Moreover, for
is an integer
and
, such that
(5)
(6)
Proof: See the Appendix.
ALON et al.: TYPICAL PEAK SIDELOBE LEVEL OF BINARY SEQUENCES
547
The next result addresses the question of how well sums of binomial coefficients can be approximated by the Gaussian complementary cumulative distribution function (CCDF). Henceforth, the Gaussian CCDF is defined by
The total number of binary sequences of length is
, therefore
Lemma 3.3: Let
Similarly, the number of sequences
, is given by
Then, for
such that
the following holdş:
(7)
and for
(8)
Therefore
(9)
Proof: See the Appendix.
We have the following lemma.
The bounds on
in Lemma 3.3 are given in terms of
. In certain cases, we would like to provide more explicit bounds, which can be achieved with the following.
Lemma 3.4: For
Lemma 4.1: Let
. For
the following holds:
Proof: Use, for
and for
holds:
It will turn out that a specific form of
following explicit bounds will be useful.
the following
is of interest. The
Lemma 3.5: For
and
Proof: Use Lemma 3.3 with
in place of .
Combining Lemmas 4.1 and 3.5, and using
, we have
Proof: See Appendix.
IV. AN UPPER BOUND ON
For
, the number of sequences
is given by
FOR
ALMOST ALL
such that
,
Consequently, we have the following corollary.
for
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 56, NO. 1, JANUARY 2010
Corollary 4.2: Under the conditions of Lemma 4.1, for
We
see
that
the
lower and upper bounds for
differ only by a multiplicative
constant.
(10)
V. A LOWER BOUND ON
Proof: Straightforward
FOR
ALMOST ALL
Notice that
are linear in
and in
, and therefore can be written collectively as a linear system
..
.
For example, taking
tain the following.
, we ob-
Corollary 4.3:
(11)
For the sake of comparison, let us derive a lower bound for
. For notational convenience, in what follows
stands for
.
..
.
..
.
This linearity allows us to prove independence of
and
for
, in Lemma 5.1. Using the
independence and the inclusion–exclusion principle, we provide
a lower bound for the upper tail on the probability of the number
with
, Theorem 5.3.
of sequences
Next, we use Azuma’s bound to show that since
satisfies a Lipschitz condition, the distribution of
is
concentrated, though we cannot indicate where its expectation
lies. However, noticing that the expectation cannot be too
small, since otherwise its upper tail—an upper bound on the
—will contradict
probability that
the earlier derived lower bound on the probability of the same
event, we conclude that the expectation cannot be less than
.
approximately
Lemma 5.1: For any
and
Lemma 4.4: For
Proof: Use Lemma 3.3 and note that for
Remark 5.2: As suggested by a referee, the key idea of the
products are independent random
statement is that
variables. A consequence is that
and
are
independent, which in turn implies the above lemma.
Proof: For
and
, we consider two forms
and
Hence
Notice that the number of product terms in the second form,
, is less than the number of product terms in the first
. Let us form a vector of length
, having
one,
the first half consisting of the first
product terms from
and the second half containing the product terms from
, namely
ALON et al.: TYPICAL PEAK SIDELOBE LEVEL OF BINARY SEQUENCES
549
Let us show that when
assumes all possible values from
, and
assumes all possible values from
, then assumes all
possible values from
equal number of times,
.
, assumes all
Notice that
possible values from
exactly once if and only if the
vector
, where stands for the coordinatewise multiplication of vectors, also assumes all possible values
exactly once. Indeed, for a fixed
asof
sumes all possible values exactly once. The same is clearly true
for a fixed
and
running over all possifor
bilities in
. In the opposite direction, the same is correct
since the transform is involution.
and
in place of
and
in the preceding
Using
expression, and noticing that the variables
do not appear in either
or
we conclude that
assumes each of its
times. Consequently,
.
for all
and
Lemma 5.4: For any
Proof: By construction
therefore, by Theorem 3.1
Applying Lemmas 3.3, 3.4 with
and
we have
possible values exactly
and
are independent
Theorem 5.3: Let
(12)
For
, we then have
Then
(15)
Proof: For any subset of sequences
subset of indices belonging to
For
and
from
, and any
Let us now return to the proof of the theorem. Note that by
symmetry
, we have
(13)
(14)
Moreover, by the inclusion–exclusion we have
For
This accomplishes the proof of Theorem 5.3.
The proof of the theorem is to be continued after we demonstrate the following auxiliary result.
Using the lower bound from Theorem 5.3 and the Azuma in.
equality, we will provide a lower bound for the mean of
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 56, NO. 1, JANUARY 2010
Lemma 5.5 (Azuma, cf. e.g., [15], [16]): Let
be
independent random variables, with taking values in a set .
satisfies, for
Assume that a function
some constants
, the following Lipschitz
differ only in the th coordinate,
condition: if two vectors
.
then
Then, the random variable
satisfies, for any
Corollary 5.6: For almost all
VI. THE
CONCENTRATION, MODULO A CONJECTURE
For notational convenience, let us introduce the following
definitions:
For
The number of sequences in a set is denoted by
.
The following idea was suggested to the authors by Alex
Koreiko.
we note that
, and therefore, for
Conjecture 6.1 (Koreiko [10]): For
Though we were not able to prove that the previous is correct,
we can show that modulo Conjecture 6.1, the following holds.
Lemma 6.2: Let
On the other hand
For
Conjecture 6.1)
, we have (modulo
For consistency, we require
Proof: As put forward by Moon and Moser in [19]
and therefore
and consequently
For
we have the inequality at the top of the following page. For
Written differently
We have thus shown that
From here, straightforward application of the Azuma inequality gives us the following.
ALON et al.: TYPICAL PEAK SIDELOBE LEVEL OF BINARY SEQUENCES
where
551
Therefore
Lemma 6.2 along with Lemma 4.4 enable us to derive a tight
lower bound for
and consequently
Corollary 6.3: For
Written differently, it is
We have thus shown that
Proof: Let
From here, the straightforward application of the Azuma inequality gives the following result.
.
Corollary 6.4: For almost all
APPENDIX I
A. Proof of Theorem 3.1
For
and we obtain the claim.
, the union bound dominates
The original proof appears in [19], here it is sketched for completeness. If
Now we may repeat the steps in the end of Section V, but this
time using Corollary 6.3, giving
there must be an excess of
Hence, the set
sets and , with
such that
Together with the Azuma inequality, it provides us with a tighter
. Indeed, the consistency
lower bound for the mean of
requirement yields
values of with
.
can be partitioned into two suband
, respectively,
if
(17)
if
(18)
and
There are
and there are
choices for the subsets
choices for the first
and
elements of
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 56, NO. 1, JANUARY 2010
. Once these choices are made, the remaining elements
are determined recursively by (17) or (18).
Since
for
B. Proof of Lemma 3.2
, we have
(24)
and any
Let us first show the upper bound. For
, we have
C. Proof of Lemma 3.3
Let us first prove (7). Let
where we have used (cf. e.g., [14])
Then
(19)
Therefore, for
and any
(20)
where
natural entropy function.
Using also
stands for the
for
Taking into account that the terms of
are monotonically
from above by the product of
decreasing, let us bound
the first (biggest) term and the number of terms in the sum, using
Lemma 3.2
(21)
(25)
and
As for
get
for
, we apply the upper bound of Lemma 3.2, to
we have
(22)
(23)
Bounding the sum with an integral, noting that for
the integrands are monotonically decreasing functions of , and recalling
Now to the lower bound. Using (19), we have for
and any
we have
Therefore, for
and any
By assumption, we have
For
, and
and
and
ALON et al.: TYPICAL PEAK SIDELOBE LEVEL OF BINARY SEQUENCES
553
Summing up and using (25)
On the other hand, if
(which can only
, i.e., for
), the summands inhappen when
and decrease thereafter. The biggest
crease for
.
summand is
can be bounded from above in both cases as
Clearly,
(26)
Noting that
(27)
under the imposed conditions
Analogously to (28), we have
(28)
We finally have
(31)
(29)
Lumping the contributions
(30)
(32)
where we have bounded
and
we get
Now, let us prove (8). Starting from the lower bound in
Lemma 3.2
where in the last inequality we used
To complete the proof, let us provide an upper bound for
Finally, noting that
Note that the maximum of
is reached for
.
If
, the summands in
are monotonically decreasing and the sum can be bounded from above by an integral
as follows:
we have
(33)
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 56, NO. 1, JANUARY 2010
D. Proof of Lemma 3.5
For
[7] S. W. Golomb and G. Gong, Signal Design for Good Correlation:
For Wireless Communication, Cryptography, and Radar. Cambridge,
U.K.: Cambridge Univ. Press, 2005.
[8] G. Halász, “On the result of Salem and Zygmund concerning random
polynomials,” Studia Scien. Math. Hung., pp. 369–377, 1973.
[9] J. Jedwab and K. Yoshida, “The peak sidelobe level of families
of binary sequences,” IEEE Trans. Inf. Theory, vol. 52, no. 5, pp.
2247–2254, May 2006.
[10] A. Koreiko, personal communication.
[11] W. Krauth and M. Mezard, “Aging without disorder on long time
scales,” Z. Phys., vol. B 97, pp. 127–131, 1995.
[12] N. Levanon and E. Mozeson, Radar Signals. Hoboken, NJ: IEEE
Press/ Wiley-Interscience, 2004.
[13] S. Litsyn and A. Shpunt, “On the distribution of Boolean function nonlinearity,” SIAM J. Discr. Math., vol. 23, no. 1, pp. 79–95, 2008.
[14] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting
Codes. Amsterdam, The Netherlands: Elsevier, 1977.
[15] C. McDiarmid, On the Methdod of Bounded Differences, ser. London
Math. Soc. Lect. Notes Ser. 141. Cambridge, U.K.: Cambridge Univ.
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[16] C. McDiarmid, Concentration. Probabilistic Moethods for Algorithmic
Discrete Mathematics. Berlin, Germany: Springer-Verlag, 1998, pp.
195–248.
[17] C. Mauduit and A. Sárközy, “On finite pseudorandom binary sequences
I. Measure, of pseudorandomness, the Legendre symbol,” Acta Arith.,
vol. 82, no. 4, pp. 365–377, 1997.
[18] I. D. Mercer, “Autocorrelations of random binary sequences,” Combin.,
Probab. and Comput., vol. 15, pp. 663–671, 2006.
[19] J. W. Moon and L. Moser, “On the correlation function of random binary sequences,” SIAM J. Appl. Math., vol. 16, no. 2, pp. 340–343,
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, we have
Noting that
we have
Further, using
for
ACKNOWLEDGMENT
The authors wish to thank the referees for insightful
comments.
REFERENCES
[1] N. Alon, Y. Kohayakawa, C. Mauduit, C. Mauduit, C. G. Moreira, and
V. Rödl, “Measures of pseudorandomness for finite sequences: Typical
values,” Proc. London Math. Soc., vol. 95, pp. 778–812, 2007.
[2] N. Alon and J. Spencer, The Probabilistic Method. New York: Wiley,
2000.
[3] J. Bernasconi, “Low autocorrelation binary sequences: statistical mechanics and configuration space analysis,” J. Physique, vol. 48, p. 559,
1987.
[4] D. Dmitriev and J. Jedwab, “Bounds on the growth rate of the peak
sidelobe level of binary sequences,” Adv. Math. Commun., vol. 1, pp.
461–475, 2007.
[5] L. Ein-Dor, I. Kanter, and W. Kinzel, “Low autocorrelated multiphase
sequences,” Phys. Rev. E, vol. 65, pp. 020102(R)-1–020102(R)-4,
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[6] S. W. Golomb, Shift Register Sequences. San Francisco, CA: HoldenDay, 1967.
Noga Alon received the Ph.D. degree in mathematics from the Hebrew University of Jerusalem, Israel, in 1983.
He is a Baumritter Professor of Mathematics and Computer Science at TelAviv University, Israel. He had visiting positions in various research institutes
including MIT, Cambridge, MA; The Institute of Advanced Study, Princeton,
NJ; IBM Almaden Center, San Jose, CA; Bell Labs, Murray Hill, NJ; Bellcore,
Morristown, NJ; and Microsoft Research, Redmond, WA. He published more
than four hundred research papers, mostly in Combinatorics and in Theoretical
Computer Science, and one book.
Dr. Alon has been a member of the Israel National Academy of Sciences since
1997 and of the Academia Europaea since 2008. He received the Erdös prize in
1989, the Feher prize in 1991, the Polya Prize in 2000, the Bruno Memorial
Award in 2001, the Landau Prize in 2005, the Goedel Prize in 2005, and the
Israel Prize in 2008.
Simon Litsyn (M’94-SM’99) was born in Kharkov, U.S.S.R., in 1957. He
received the M.Sc. degree from Perm Polytechnical Institute, Perm, U.S.S.R.,
in 1979, and the Ph.D. degree from Leningrad Electrotechnical Institute,
Leningrad, U.S.S.R., in 1982, both in electrical engineering.
Since 1991, he has been with the Department of Electrical Engineering–Systems, Tel-Aviv University, Israel, where he is a Professor. His research interests
include coding and information theory, communications, and applications of
discrete mathematics. He authored Covering Codes (Elsevier, 1997) and Peak
Power Control in Multicarrier Communications (Cambridge University Press,
2007).
Dr. Litsyn received the Guastallo Fellowship in 1992. During 2000–2003, he
served as an Associate Editor for Coding Theory for the IEEE TRANSACTIONS
ON INFORMATION THEORY. He is an editorial board member of Advances
in Mathematics and Communications (AMC) and Applicable Algebra in
Engineering Communication and Computing (AAECC).
Alexander Shpunt received the B.Sc. degree in physics/computer science in
1999, followed by the M.Sc. degree in electrical engineering in 2005, both with
high honors, from Tel-Aviv University, Tel-Aviv, Israel. In 2006, he moved to
the Department of Physics at the Massachusetts Institute of Technology, Cambridge, where he currently is a fourth year Graduate Fellow.